manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.
A hyperbolic manifold is a geodesically complete Riemannian manifold $(X,g)$ of constant sectional curvature $-1$.
Of particular interest are hyperbolic 3-manifolds.
Every hyperbolic manifold is a conformally flat manifold.
(e.g. Long-Reid 00, p. 4)
There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.
The Mostow rigidity theorem states that every hyperbolic manifold of dimension $\geq 3$ and of finite volume is uniquely determined by its fundamental group.
A Riemannian manifold
with zero sectional curvature is a Euclidean manifold?;
with +1 sectional curvature is an elliptic manifold?
See also
Textbook accounts:
John Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (doi:10.1007/978-0-387-47322-2, pdf)
Michael Kapovich, Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics, Birkhäuser 2008 (doi:10.1007/978-0-8176-4913-5)
See also
Darren D. Long, A. W. Reid, On the geometric boundaries of hyperbolic 4-manifolds (arXiv:math/0007197)
Wikipedia, Hyperbolic manifold
Wikipedia, Hyperbolic space
Last revised on July 21, 2020 at 17:51:36. See the history of this page for a list of all contributions to it.